On the simultaneous restoration of the density and the surface equation in the inverse gravimetry problem for a contact surface

Analytical and numerical methods for solving inverse problems of logarithmic and the Newtonian potentials are investigated. The following contact problem in the case of a Newtonian potential is considered. In the domain $\Omega\{\Omega: -l\leqslant x,y\leqslant l, H-\varphi(x,y)\leqslant z\leqslant H\}$, sources with the density $\rho(x,y)$, perturbing the Earth's gravitational field, are distributed. Here, $\varphi(x, y)$ is a non-negative finite function with the support $\Omega=[-l,l]^2$, $0\leqslant\varphi(x,y)\leqslant H$. It is required to simultaneously restore the depth $H$ of the occurrence of the contact surface $z=H$, the density $\rho(x,y)$ of sources, and the function $\varphi(x,y)$. The methods of simultaneous determination are based on the use of nonlinear models of potential theory which are developed in the paper. The following kinds of information are used as the basic ones: 1) values of the gravity field and its first and second derivatives; 2) values of the gravity field at the different heights. The possibility of the simultaneous recovery of the functions $\rho(x,y)$, $\varphi(x,y)$ and the constants $H$ in the analytical form is demonstrated. Iterative methods for their simultaneous recovery. The model examples demonstrate the effectiveness of the proposed numerical methods are constructed.